Systematics ofc-axis phonons in the thallium- and bismuth-based cuprate superconductors

We present grazing incidence reflectivity measurements in the far-infrared region at temperatures above and below $T_c$ for a series of thallium- $({\mathrm{Tl}}_2{\mathrm{Ba}}_2{\mathrm{CuO}}_6,$ ${\mathrm{Tl}}_2{\mathrm{Ba}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8)$ and bismuth- $({\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CuO}}_6,$ ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8,$ and ${\mathrm{Bi}}_{2-x}{\mathrm{Pb}}_x{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8)$ based cuprate superconductors. From the spectra, which are dominated by the $c$-axis phonons, longitudinal optical (LO) frequencies are directly obtained. The reflectivity curves are well fitted by a series of Lorentz oscillators. In this way the transverse optical (TO) phonon frequencies were accurately determined. On the basis of the comparative study of the Bi- and Tl-based cuprates with a different number of ${\mathrm{CuO}}_2$ layers per unit cell, we suggest modifications of the assignment of the main oxygen modes. We compare the LO frequencies in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8$ and ${\mathrm{Tl}}_2{\mathrm{Ba}}_2{\mathrm{Ca}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{10}$ obtained from intrinsic Josephson junction characteristics with our measurements, and explain the discrepancy in LO frequencies obtained by the two different methods.